Secondary Flow and Upstream Dynamics in Double
Bifurcation Model
Fong Yew Leonga, Kenneth A. Smitha,b, Chi-Hwa Wanga,c
a
Singapore-MIT Alliance, b Massachusetts Institute of Technology, c National University of Singapore
Abstract – Flow behavior in bifurcation models is of
great importance to health risk assessments and
pulmonary drug delivery. This is particularly true
of secondary flow behavior in multi-bifurcation
models.
Previously,
both
numerical
and
experimental methods have shown that four-vortex
secondary flow structures can develop in the crosssections of grand-daughter branches. This work
shows that the development of secondary flow in
the grand-daughter tubes is due to local stretching
of vortex lines in the upstream DT. Scaling
arguments have been used to derive two critical
parameters governing this particular vorticity
transport problem. A simple model for vorticity
generation and transport is proposed, taking into
account the geometric limitations imposed by the
rigid walls of the tubes.
Index Terms – bifurcation, lung, secondary flow,
vortex stretching
I. INTRODUCTION
The secondary flows produced by bifurcations are of
great importance to an understanding of aerosol
transport to lung tissue and pulmonary drug delivery.
Steady flow in single bifurcation models yields
counter-rotating vortices in the daughter branches
during the inspiratory phase [1]. These are
qualitatively not unlike those commonly found in flow
in curved tubes [2,3]. The complexity of the
bifurcation geometry almost demands models to be
based on computational fluid dynamics (CFD),
although some insights can be gained from semianalytical works involving a mother tube with circular
cross-section bifurcating into semi-circular daughter
tubes[4,5]. In human airways, the typical length-toThis work is supported by the Singapore-MIT Alliance
Fong Yew Leong is with the Singapore-MIT Alliance,
MEBCS Program, National University of Singapore, 4
Engineering Drive 3, Singapore 117576, Singapore
Kenneth A. Smith is with the Singapore-MIT Alliance and the
Massachusetts Institute of Technology, 77 Massachusetts Ave
Cambridge, MA 02139, USA
Chi-Hwa Wang is with the Singapore-MIT Alliance and the
Department of Chemical and Biomolecular Engineering,
National University of Singapore, 10 Kent Ridge Crescent,
Singapore 119260, Singapore
diameter ratios do not allow for fully developed flow
profiles between bifurcations, as shown by Weibel’s
morphometric lung model [6].
Recent CFD efforts have revealed more complicated
flow structures in symmetric double bifurcation
models [7,8] and triple bifurcation models [9,10].
These symmetric bifurcation models may not be fully
representative of actual lung casts, but are frequently
popular for direct comparison of experimental and
theoretical findings. There are also multi-bifurcation
models based on physiologically realistic lung
geometries [11,12]. These models may be more
physiologically representative but they also tend to
yield more complicated solutions which may mask the
underlying physical phenomena.
Multi-bifurcation models have revealed complex
secondary flows with multi-vortex configurations. In a
double bifurcation model for instance, the prevalent
secondary flow structure in the grand-daughter
branches is shown to be a set of four counter-rotating
vortices at high Reynolds numbers [11].
The
additional set of counter-rotating vortices close to the
inner walls of the grand-daughter branches are
opposite in sense to those found in the case of the
centrifugally-induced vortices observed in single
bifurcation models. There is, however, a lack of
detailed understanding in the development of these
vortices in multi-bifurcation models despite the various
numerical solutions to physiologically representative
bifurcation models.
In addition to numerical simulations, there are a
number of experimental studies which report the
transient flow behavior of fluids in rigid single
bifurcation models. Although these experimental
models lack anatomical details of the human airways,
they can be reliably reproduced for validation purposes
[7]. Earlier works involve velocity measurements using
the hot-wire probes and smoke tracers for flow
visualization [13]. Flow visualization in single
bifurcation models has earlier revealed not only the set
of counter-rotating vortices in the secondary flow of
the daughter branches, but also a characteristic double
peaked axial velocity profile in the plane of the
bifurcation and a characteristic M-shaped profile in the
plane perpendicular to the bifurcation. Modern
measurement techniques have included Laser-Doppler
Velocimetry (LDV) and Particle Image Velocimetry
(PIV). These non-invasive optical techniques often
yield good resolution on primary axial [14]. The
difficulty posed by secondary flows is highlighted in
[15], which cites diffraction losses and imaging depths
as major sources of measurement errors. To the
knowledge of the authors, there is as yet no reported
experimental work on secondary flow in a double
bifurcation.
As an illustrative double bifurcation model, here we
present the dependence of the secondary flow behavior
in the grand-daughter branches on the upstream
conditions and the geometry of the carinal ridge. The
problem is handled through both numerical and
experimental approaches.
II. MATERIAL/VORTEX LINES
In general, the characteristic Re has to be much larger
than unity in order to yield the four-vortex
configuration in the grand-daughter tubes. It is
therefore anticipated that the inviscid assumption will
be valid except in the near-wall region. Therefore, the
vortex line will always contain the same fluid particles.
We can in fact exploit this classical feature by tracing
the evolution of vortex lines connected by a single
velocity streamline.
Starting with the RGDT of the double bifurcation
model (Re 250), we first specify two reference
positions corresponding to the vortex centers in the
upper symmetrical half of an arbitrary cross section in
the RGDT (one that intersects the carinal ridge is
chosen for this study). We first trace a velocity
streamline from the reference position at the inner
vortex (one near the inner wall of the GDT), and
several locus points are identified on the streamline.
Vortex lines are now traced from those loci as shown
in Fig. 1. These vortex line are in fact closed (as
expected), and so for simplicity, they are termed as
vortex rings (note the labels VR-A to VR-D). The
color scheme refers to the dimensionless vorticity in
the component parallel to the RGDT (positive in the
direction towards the outlet). The top-down view is
provided in Fig. 1a, and the side-view perspectives are
shown in Fig. 1b and Fig 1c.
The upstream initial condition for the vortex ring is
represented by VR-A which is a half-ring due to the
symmetry condition at the MT. It carries null vorticity
in the direction parallel to the RGDT but it is not
irrotational. At VR-B, the half-ring from VR-A is
joined by another half-ring near the wall. The original
half-ring is noticeably extended in the direction of
RGDT and the vortex ring becomes curved (Fig. 1c).
The actual stretching in fact takes place at four
localized regions on the vortex ring: two on each halfring. This results in an intensification of vorticity in the
component parallel to RGDT at those specific regions.
At VR-C, the vortex ring has adopted a C-shaped form,
which corresponds well with the commonly observed
‘M-shaped’ axial velocity profiles in single
bifurcations. Just before the vortex ring reaches the
carinal ridge at VR-D, the original half-ring now
manifests as the inner vortices at the inner wall of
RGDT (Fig. 1c). In contrast, the other half-ring
generated from the DT inner wall contributes to the
formation of the outer vortices. Hence, we deduce that
the basis for vortex formation in the GDT is already
established in the upstream DT through the extension
of the vortex lines.
Similar method has been used for the LGDT, starting
from plotting velocity streamline from a specified
reference point on the inner vortex core, followed by
tracing four vortex rings from selected points on the
streamline. These vortex rings, which are identified as
VR-E, VR-F, VR-G and VR-H, are shown in the set of
figures on Fig. 2. We observe similar qualitative
features between the cases for LGDT and RGDT. An
illustrative upstream initial condition is presented by
VR-E, which is a half-ring due to the symmetry
condition but carries finite vorticity in the direction
parallel to the LGDT as the result of the fully
developed entrance profile (in contrast with VR-A).
The vortex ring VR-F is now a full ring, which curves
inwards to a C-shaped VR-G. Finally, interaction with
the carinal ridge deforms the vortex ring and it is now
‘butterfly-shaped’ (quad-lobed) as shown by VR-H.
The extension of vortex line in the direction of the
LGDT has resulted in an increased vorticity in the
LGDT component at four localized regions (Fig. 2b).
This has led to the formation and observation of the
four-vortex structure in the LGDT.
Fig. 1a (Top): Top-down view of double-bifurcation
model (Re 250). Vortex rings linked by velocity
streamline through inner vortex (RGDT). Colorbar shows dimensionless vorticity in RGDT
component. Fig. 1b (Middle): Side-view 1; Fig. 1c
(Bottom): Side-view 2
Fig. 2a (Top): Top-down view of double-bifurcation
model (Re 250). Vortex rings linked by velocity
streamline through inner vortex (LGDT). Colorbar shows dimensionless vorticity in LGDT
component. Fig. 2b (Middle): Side-view 1; Fig. 2c
(Bottom): Side-view 2
III. SCALING LAWS
The numerical simulation in the previous section has
shown that localized extension of the vortex lines in
the upstream daughter tube is responsible for the
generation of vorticities in the grand-daughter tubes.
This finding suggested that it may be possible to have
a simple physical explanation for the secondary flow
structures in the grand-daughter tubes.
We first begin with the vorticity transport equation.
∂ω
(1)
+ u ⋅ ∇ω = ω ⋅ ∇ u + ν ∇ 2 ω
∂t
For steady state analysis, the temporal term on the left
can be neglected, and that leaves us with the remaining
three other terms to work on. Even though there are
three scalar component equations of the vorticity
transport vector equation (1), we can think of the flow
as consisting of the main flow along of the axis of the
tube (axial component) and the secondary flow
perpendicular to the axis (secondary component),
which are denoted as x and y respectively.
In this way, the convective term in the axial and
secondary directions can be described respectively as
∂ω y
∂ω y
∂ω x
∂ω x
(2)
u
∂x
+v
∂y
u
∂x
+v
Similarly, the vortex stretching term can be written as
∂v
∂v
∂u
∂u
(3)
ωx + ω y
ωx
+ ωy
∂x
∂y
∂x
∂y
And finally, the viscous term can be written as
∂ 2ω x
∂2ω
(4)
ν
ν 2y
2
∂x
∂y
where ν is the kinematic viscosity of the fluid.
If we consider solely the axial component of the
vorticity, we have estimates for the inertial term as
∂ω x UΩ x
(5)
∂x
~
R
The secondary flow problem is now considered. Based
on what has been observed in the numerical simulation
previously, the vortex line (or ring) is distorted and
stretched by the secondary flow as shown in the
schematic diagram (Fig. 3). This in turn leads to an
increase in the vorticities along the vortex line with
respect to the grand-daughter tubes (GDT).
Hence it is observed that in the absence of secondary
flow, there can be no stretching of the vortex line in
the cross-section of DT. In the case of the bifurcation
geometry, the secondary flow is generated by the
centrifugal acceleration, which is caused by the fluid
having to negotiate a curvature. Hence, it is concluded
that secondary flow is insignificant without centrifugal
acceleration (it is assumed that displacement effect
caused by new intervening wall surface at the carinal
ridge is negligible).
Vortex line
Vortex line
∂y
where u and v are the components of the velocity
vector in the axial (x) and secondary (y) directions; and
ωx and ωy are the respective components of the
vorticity vector.
u
flow acting through the pressure gradient term in the
momentum equations (which vanished as a term when
the curl of the equation was taken). A representative
term for the centrifugal acceleration would then be
u 2 , where R is the radius of curvature.
L
And the viscous term can be scaled as
∂ 2ω x νΩ x
(6)
~ 2
ν
∂y 2
δ
Where δ is an unspecified viscous length-scale (or
boundary layer thickness)
It is worth noting that a bifurcation, in the absence of
curvature, is no different from a straight tube. In other
words, it is the centrifugal acceleration due to the
curved bifurcation geometry that drives the secondary
Outer
wall
Inner
wall
Outer
wall
Inner
wall
X=0
X=L
Fig. 3 (Left): Schematic Diagram of Vortex Line in
Cross-section of DT. Fig. 3 (Right): Vortex Line at
an arbitrary distance L downstream. Dashed lines
refer to secondary flow streamlines. Bold red lines
refer vortex lines. Shaded region refers to a local
increase in secondary vorticity
Hence, we examine the secondary flow equation of
motion and gather the order of magnitude estimates in
the following manner:
Convective: u ∂v v ∂v
∂x ∂y
Centrifugal:
u2
R
2
Viscous: ν ∂ v
∂y
2
2
Estimate: UV V
L a
Estimate: U
2
(7)
(8)
R
Estimate:
νV
δ2
(9)
Since the scales for the convective and viscous terms
(7 and 9) are functionally dependent on V, we start a
hypothetical scaling argument about the centrifugal
acceleration term.
• If the axial convective term is dominant, then
2
UV U =>
~
L
R
V ~
L
U
R
(10)
This would require the axial convective term to be
dominant over both the radial convective term and
the viscous term. In other words, the following
inequalities must be satisfied, using the scaling for
V determined in (10).
2
V 2 => ⎛ L ⎞ ⎛ a ⎞ << 1
UV
(11)
>>
⎜ ⎟ ⎜ ⎟
L
a
⎝a⎠ ⎝R⎠
UV
νV => ⎛ Uδ ⎞⎛ δ ⎞
(12)
>> 2
⎜
⎟⎜ ⎟ >> 1
L
δ
⎝ ν ⎠⎝ L ⎠
• If the secondary convective term is dominant, then
⎛a⎞
V 2 U 2 =>
V ~⎜ ⎟
~
L
R
⎝R⎠
U
1/ 2
V2
UV
>>
a
L
=> ⎛⎜ L ⎞⎟⎛⎜ a ⎞⎟
V2
νV
>> 2
a
δ
=> ⎛ a ⎞ ⎛ Uδ ⎞⎛ δ ⎞ >> 1
>> 1
1/ 2
⎜ ⎟
⎝R⎠
⎟⎜ ⎟
⎜
⎝ ν ⎠⎝ a ⎠
2
(14)
(15)
⎛ Uδ ⎞ ⎛ δ ⎞⎛ δ ⎞
⎜
⎟ ⎜ ⎟⎜ ⎟ << 1
⎝ ν ⎠ ⎝ R ⎠⎝ a ⎠
(18)
Based on the inequalities specified above, two key
parameters are identified, namely ⎛ U δ ⎞ ⎛ δ ⎞ and
⎜
⎟⎜ ⎟
⎝ ν ⎠⎝ L ⎠
⎛ L ⎞⎛ a ⎞
⎜ ⎟⎜ ⎟
⎝ a ⎠⎝ R ⎠
1/ 2
namely
⎛ Uδ ⎞ ⎛ δ ⎞
⎜
⎟⎜ ⎟
⎝ ν ⎠⎝ L ⎠
and ⎛⎜ L ⎞⎟⎛⎜ a ⎞⎟
1/ 2
, a map of dominant
⎝ a ⎠⎝ R ⎠
regimes for the vorticity transport equation can be
constructed as shown in Fig. 4.
⎛ L ⎞⎛ a ⎞
⎜ ⎟⎜ ⎟
⎝ a ⎠⎝ R ⎠
⎛ Uδ ⎞ ⎛ δ ⎞
⎜
⎟⎜ ⎟
⎝ ν ⎠⎝ L ⎠
1/ 2
I – Axial
Convective
~1
IIa –Stretching
(Convective)
⎛ Uδ ⎞⎛ δ ⎞
⎜
⎟⎜ ⎟ ~ 1
⎝ ν ⎠⎝ L ⎠
(13)
• Lastly, if the secondary viscous term is dominant,
νV U 2 =>
⎛ Uδ ⎞⎛ δ ⎞
(16)
~
V ~⎜
⎟⎜ ⎟U
R
δ2
⎝ ν ⎠⎝ R ⎠
Similarly, this would require the viscous term to be
dominant over both the convective terms, in the
spirit of the boundary layer theory. The following
inequalities must be satisfied, using the scaling for
V determined in (16).
νV
UV => ⎛ Uδ ⎞⎛ δ ⎞ << 1
(17)
⎜
⎟⎜ ⎟
>>
L
δ2
⎝ ν ⎠⎝ L ⎠
νV
V 2 =>
>>
2
a
δ
Based on the two parameters as identified earlier,
1/ 2
This would require the secondary convective term
to be dominant over both the axial convective term
and the viscous term. The following inequalities
must be satisfied, using the scaling for V
determined in (13).
⎝ a ⎠⎝ R ⎠
equation of motion, it does not affect the dominance of
the respective terms in the vorticity transport equation.
, which are used to scale the secondary flow.
This allows us to define three separate transport
regimes (axial convective, radial convective and
viscous).
Hence, the scaling argument for the vorticity equation
is exactly identical to the equation of motion (since all
the terms scale by a constant). The corollary to this
observation is that, even though the scale for V
depends on the dominance of respective terms in the
IIb – Stretching
(Viscous)
III - Viscous
⎛ U δ ⎞⎛ δ ⎞ ⎛ a ⎞
⎜
⎟⎜ ⎟ ⎜ ⎟
⎝ ν ⎠⎝ a ⎠ ⎝ R ⎠
1/ 2
~1
⎛ L ⎞⎛ a ⎞
⎜ ⎟⎜ ⎟
⎝ a ⎠⎝ R ⎠
1/ 2
Fig. 4: Schematic Diagram of Flow Regimes Based
on Scaling Arguments
IV. VORTICITY TRANSPORT MODELING
Since the earlier scaling argument can only provide an
order-of-magnitude description of the problem, we
seek to improve the accuracy of the theoretical
estimates. The following discussion is based on the
premise that vortex stretching is the dominant source
1/ 2
of secondary vorticity, or in other words, ⎛ L ⎞⎛ a ⎞ >> 1 .
⎜ ⎟⎜ ⎟
⎝ a ⎠⎝ R ⎠
With that in mind, hypothetical transport models with
varying geometrical assumptions are proposed.
We shall begin from the solution of the simplest case
without the vortex stretching term (i.e. pure
convective-diffusive transport) before moving on to
other cases which involves the analysis of the vortex
stretching term.
Case One: Pure Convective-Diffusive Transport
It is assumed that the generation of secondary vorticity
occurs instantaneously over a very short time-scale, so
that the secondary vorticity generation profile
resembles a Dirac delta. This allows us to neglect the
vortex stretching term from the vorticity transport
equation (since the bulk of the transport is convective-
diffusive). The simplification is substantial because
vorticity only enters the problem as a transportable
scalar quantity to be specified as an initial condition.
The boundary
vorticity ω y are
ω y ( x,0) = 0
(21)
For a two-dimensional case as depicted in Fig. 5, we
are examining fluid flow at relatively high Reynolds
numbers between a pair of parallel plates, which
corresponds to the upper and lower walls of the
daughter tube. The coordinates of interest here are the
horizontal x and the vertical z. The assumed uniform
velocity condition far upstream (x < 0) encounters a
sudden curvature at the entrance (x = 0) and then
straightens out for a distance of 0 < x < L. It is
assumed that the fluid instantaneously attains an
unknown quantity of secondary vorticity, which we set
ω y ( x, a ) = 0
(22)
ω y (0, z ) = Ω y ( z )
(23)
as
Ω y (z )
for the initial vorticity condition (the
subscript y is the out of plane vector orthogonal to x
and z). The increase in vorticity in this twodimensional problem modifies the velocity gradient so
the velocity profile deforms as shown in Fig. 5, where
the vorticity on the symmetrical top and bottom halves
are opposite in directions. As the fluid moves towards
the outlet (x = L), the outward diffusion of vorticity
causes the velocity gradient to decay. The preceding
description of the problem resembles strongly the case
of the wake of the flow past a flat plate.
conditions
a
Axial centerline
With the boundary conditions (21-23), the boundary
value problem is now fully specified and sufficiently
simple to solve by a variety of analytical methods. The
confined dimensional specification of the problem
suggests that the similarity method is not the best
method, so instead the Finite Fourier Transform (FFT)
method is used. Detailed solution is provided in
Appendix A.
U
x=0
vorticity impulse due to sharp
curvature (out of plane)
⎛
n =1
⎝
Ω yn = ω yn (0) = ∫
1
(
(nπ )2 x ⎛ L ⎞ ⎞⎟ sin nπz (24)
⎜ ⎟
Re a ⎝ a ⎠ ⎟⎠
)
2 sin nπz Ω y ( z )dz
where the variables on overbars indicate that they have
been rendered dimensionless via ω y = ω y (a / U ) ,
x=L
x = x / L and z = z / a
Fig. 5: Schematic Diagram of Flow Between Two
Parallel Plates With Specified Upstream Vorticity
We define velocity difference as
∧ ( x, z ) ≡ U − u ( x, z )
∞
0
x
secondary
of the velocity profile ( ∂u / ∂z = 0 ) is positioned at
z ≅ a , where 0 ≤ z ≤ a . The last boundary condition
(23) is an entrance condition and also the upper limit
of the vorticity without further vortex stretching.
ω y ( x, z ) = 2 ∑ Ω yn exp⎜⎜ −
z
the
The first boundary condition (21) is based on the
condition of null shear stress at the plane of symmetry
(z = 0). The second boundary condition (22) requires
the assumption that the boundary layer thickness of the
upper wall is thin ( δv <<a), so that the stationary point
The solution obtained is
Wall
for
(19)
It is assumed that ∧ << U , which at low vorticity
magnitudes, may in fact be easier to satisfy in this
present problem than for the case of wake past a flat
plate. The reason is that the centerline velocity at the
origin does not have to be zero, as is the case for a flat
plate (due to the enforced no-slip condition).
Taking the curl of the momentum equation and
simplifying, we obtain
∂ω y
∂ 2ω y
(20)
=ν
U
∂x
∂z 2
We first draw our attention to the exponential factor. It
basically tells us that
• The parameter of importance in the axial
transport of vorticity is Re a−1 ⎛⎜ L ⎞⎟ . This is
⎝a⎠
certainly related to the boundary layer
parameter ⎛ Uδ ⎞⎛ δ ⎞ that we have obtained
⎜
⎟⎜ ⎟
⎝ ν ⎠⎝ L ⎠
earlier in the derivation of the scaling laws for
the case of δ = a .
•
Secondary
vorticity
decays
almost
exponentially with downstream axial distance.
This result provides an insight into how the
vorticity quantity is transported and was not
obtained directly in the previous scaling
arguments. It may be helpful in the
comparison with numerically obtained values.
Case Two: Generation of Secondary Vorticity and
Convective Transport
In the attempt to resolve the convective-diffusive
development of secondary by analytical means, the
generation of secondary vorticity in Case One had
been neglected.
How is secondary vorticity generated in the daughter
tube? It is known that the pair of counter-rotating
vortical secondary flow structure found in daughter
tubes is mainly caused by the centrifugal acceleration
of the fluid having to negotiate a segment of finite
curvature. Even though these vortices are
predominantly axial in direction, the fluid shear
distorts and extends the vortex lines along the crosssectional plane, thus increases the secondary vorticity
by stretching (see Fig. 6). As suggested in the earlier
scaling arguments, it is assumed that the secondary
vorticities should scale similarly to the secondary flow
(due to the similarity between the secondary equation
of motion and the vorticity transport equation).
z
Vortex line at X = 0
Vortex line
at X = L
Secondary flow
streamline
2Z’
Z’
θ
Y’
Secondary
flow
streamline
Symmetry line
Symmetry line
y
Fig. 6: Schematic Diagram of Vortex Lines in
Cross-section of DT (Left) and the characteristic
secondary flow streamline (Right)
Close examination of the scaled secondary flow
1/ 2
parameter ⎛⎜ L ⎞⎟⎛⎜ a ⎞⎟ suggests that it is not sufficient
⎝ a ⎠⎝ R ⎠
to use this parameter directly in the analysis of the
secondary vorticity. Instead, it is noted that the crosssectional geometry of the daughter tube can in fact
impose a physical constraint on the growth of the
secondary vorticity. More specifically, the vortex line
cannot possibly stretch indefinitely in any secondary
direction, due to the no-penetration condition of the
side walls.
characteristic secondary flow streamline is defined
with the center positioned at the center of the upper
half of the cross-section, or a distance of Z ' = a / 2 ,
where a is the radius of the tube. The characteristic
minor semi-axis of the ellipse is Z’ and the
characteristic major semi-axis is Y’, where
Y ' = 2 Z ' = 2a / 2 .
The characteristic angular velocity along the secondary
flow streamline is defined as V, which we can relate to
our previous scaling argument. We assume that
secondary convection due to centrifugal acceleration is
significant, so that the scale for secondary velocity is
defined as V ~ ⎛ a ⎞1 / 2 U (13), where R is the radius of
⎜ ⎟
⎝R⎠
curvature and U is the characteristic axial velocity.
We are interested in the stretching of vortex line in the
y-component, hence we start by resolving the
secondary velocity in the y-component. The ellipse as
shown in Fig. 6 (Right) can be expressed in polar
coordinates as
y = −Y ' sin θ
z = Z ' (1 + cos θ )
(25)
We shall restrict our analysis to a parametric angle of
0 ≤ θ ≤ π / 2 (Fig. 6 Right). The time scale parameter
is defined as t ~ x / U where x is the axial component.
The derivative of y with respect to time t (in subscripts)
is written as
(26)
yt = −Y ' cosθ ⋅ θ t
Now the angular velocity can be expressed as a
function of the secondary velocity θ t = V
r
Since r = y 2 + ( z − Z ')2 , eqn (26) can be written as a
function of time t only:
yt
=
V
=
Y ' cosθ
y + ( z − Z ')
2
2
(27)
(Y ' cosθ )2
(Y ' sin θ )2 + (Z ' cosθ )2
2
⎛
⎛ Z'⎞ ⎞
2
= ⎜ (tan θ ) + ⎜ ⎟ ⎟
⎜
⎝ Y ' ⎠ ⎟⎠
⎝
−1 2
t
It appears to be impossible to obtain an exact solution
to the vortex stretching process without solving the full
momentum equations. Instead, we propose a simplified
working model.
A schematic diagram of the upper symmetrical half of
the cylinder cross-section is as shown in Fig. 6. In this
model, several assumptions are made. First of all, a
Where θ = V dt
∫
0
r
From eqn (27), we verify that
yt
⎛Y'⎞
→⎜ ⎟
V
⎝ Z'⎠
as
θ →0
(28)
yt
→0
V
θ →π /2
as
(29)
What the above limits tells us is that the rate of vortex
extension starts off at the theoretical maximum as
indicated in (28) and decreases to zero as the
trigonometric limiting term becomes significant as
indicated in (29). This is important because it is
indicative of the limitations of the generation of
secondary vorticity via vortex stretching.
With the assumed secondary velocity scale of
, we generalize an estimation of the vortex
⎛a⎞
1/ 2
V ~⎜ ⎟ U
⎝R⎠
stretching term as
⎛
yt ⎡
⎛ Z'⎞
2
~ ⎢(aR )⎜ (tan θ ) + ⎜ ⎟
⎜
a ⎢⎣
⎝ Y' ⎠
⎝
2
⎞⎤
⎟⎥
⎟⎥
⎠⎦
−1 2
(30)
U
For this assumed case with negligible viscous effects,
we can simplify the vorticity transport equation in the
same way as was shown in Case 1.
2
⎡
⎛
⎛ Z ' ⎞ ⎞⎤
2
≈ ⎢(aR )⎜ (tan θ ) + ⎜ ⎟ ⎟⎥
⎜
∂x
⎝ Y ' ⎠ ⎟⎠⎥⎦
⎢⎣
⎝
∂ω y
−1 2
(ω )
(31)
y
We assume that the radius of curvature depends
weakly on x. For the case of θ → 0 which occurs
near the bifurcation point x → 0 , eqn (31) simplifies
to
∂ ln ω y
1 Y'
(32)
x →0 ≈
∂x
aR Z '
which implies exponential dependence of the
secondary vorticity on early axial distance.
V. NUMERICAL ANALYSIS
Numerical study has been conducted for an
incompressible fluid flow in a planar double
bifurcation model with a variable length-to-diameter
ratio in the DT. The objective is to analyze the effects
of the axial length-scale on vorticity transport and
verify the hypothesized vorticity transport model. All
other geometric parameters are held constant, e.g. the
bifurcation angles for both first and second
bifurcations are set at 90º. Boundary conditions are
fully developed flow at the inlet, no-slip condition at
the walls and null pressure condition at the outlets.
Numerical simulations are conducted under quasisteady conditions for a range of Re up to 336, using a
sinusoidal varying velocity input. A total of 20
different sets of double bifurcation models are
examined, each differing from one another only by the
length of the DT, measured from bifurcation center to
bifurcation center. Numerical convergence for each
case required approximately 24 hours, and the number
of finite elements used ranges from approximately
40,000 to 80,000. With the converged solution field, a
cross-section is taken from LGDT and RGDT as
shown in Fig. 7, each one provided an allowance of a
fixed distance of 1.5a from the second bifurcation axis
(where a is the radius of the MT and the characteristic
length-scale of the problem). We shall define a length
parameter here as dimensionless axial distance, which
is the sum of the length of DT and the fixed distance,
normalized by the characteristic length a.
On the other hand, for θ → π / 2 which occurs at
some unknown axial distance x = L’, eqn (32)
simplifies to
∂ω y
(33)
→0
∂x
x →L '
Hence, we recognize that in the absence of viscous
effects, a plot of secondary vorticity as a function of
axial distance in fact suggests a sigmoidal trend, i.e. an
early exponential growth followed by decreasing
growth rates and eventually a stationary point.
Subsequently, at larger values of axial distance x,
vorticity generation due to vortex stretching becomes
insignificant and the viscous effects become dominant.
In this case we revert to the earlier presented results for
Case 1.
Fig. 7: Schematic Diagram of Double Bifurcation
Model (Symmetrical about centerline of MT) –
Length of DT is varied in each simulation run
In each cross-section, the vorticity field in the direction
of the respective GDT can be derived. Thus the
maximum point vorticity magnitudes can be identified
by inspection. Since the objective of this study is to
analyze the vorticity due to the stretching of vortex
lines in the fluid core, the near-wall vorticities
magnitudes are neglected. In each cross-section under
inspection, either two or four distinct regions of
elevated vorticity magnitudes can be identified.
Only the vorticity component in the direction of each
respective GDT is considered. For convenience, this
variable is loosely termed as ‘streamwise vorticity’ in
the subsequent analysis, even though the local primary
velocity may not exactly parallel to the respectively
GDT. This streamwise vorticity ωGDT is rendered
dimensionless via ω = ωGDT ⋅ a , where a is the radius
GDT
U
of MT and U is the centerline velocity of the MT.
Fig. 8 shows the variation of the averaged maximum
streamwise vorticity magnitudes for the four different
vortex regions under the effect of varying
dimensionless axial distance, at an instantaneous Re of
336.
We observe that the profiles for the inner and outer
vorticities are distinctive and different. The outer
vorticity for the RGDT increases sharply for small
values of dimensionless axial distance, and
subsequently declines with increasing values of that
parameter. On the other hand, the outer vorticity for
the LGDT increases monotonically with increasing
distance within the range covered by this study. The
inner vorticity profiles for both GDT are qualitatively
similar, with an increase in magnitude, until the axial
distance of about 4 is reached, and later decays with
increasing axial distance. Also note the cross-over
intersecting point between the LGDT vorticities at an
axial distance of approximately 6.2.
Since only the inner vorticities originate from the DT
fluid core and are accompanied by stretching of the
vortex line, we will be looking at these in greater detail.
In the schematic shown in Fig. C earlier, the early
increase in vorticities is hypothesized to be exponential
in the form of eqn (32), which can be expressed as
⎛Y'
ω y ~ exp⎜⎜
⎝ Z'
a⎞
⎟x
R ⎟⎠
initial data points both inner vortex curves reveals an
average exponential value of about 2.6 (as shown in
Fig. 8). Then, matching the regression profile and eqn
(34), and assuming a value of 1.5 for the ratio of the
semi-axes (Y’/Z’) of the hypothetical ellipse, the
estimate for the curvature ratio (a/R) is about 0.33,
which seems to be a reasonable figure.
From eqn (33)
∂ω y
∂x
x →L '
→0
(35)
There is not much to be done about eqn (35), since it
only predicts a stationary point in the vorticity profile,
and it is already observed to be the case as shown in
Fig. 8.
From eqn (24)
∞
⎛
ω y ( x , z ) = 2 ∑ Ω yn exp⎜⎜ −
(nπ )2 x ⎞⎟ sin nπz
(36)
Re a ⎟⎠
⎝
The final eqn (36) suggests an exponential decline of
the vorticity with increasing axial distance. The first
term Regression of the terminal data points of both
inner vortex curves reveals exponential values of -0.33
and -0.49 for the RGDT and LGDT respectively. This
is approximately one order of magnitude higher than
the predicted value of ~0.04. This suggests that
vorticity decay may not be the only explanation for the
decline of vorticity at large axial lengths.
n =1
Reverting to the vorticity transport equation, it appears
probable that the explanation for the observed
phenomenon is the presence of a non-trivial vortex
destretching term. Unlike vortex stretching, it is not
immediately clear how the destretching process can
occur. It is speculated that viscous decay of the
secondary flow velocity results in a negative
acceleration in the secondary direction. Unfortunately,
the inclusion of the vortex stretching term makes the
vorticity transport equation highly non-linear, so this
particular issue has not been resolved as yet.
(34)
Where Y’ and Z’ are the major and minor semi-axes of
the hypothetical ellipse as shown in Fig. 4.2 (Right), a
is the radius of MT, R is the radius of curvature and x
is axial distance. We assume that the actual profile is
indeed exponential as predicted and regression of the
Location
RGDT
RGDT
LGDT
LGDT
Inner
Outer
Inner
Outer
Vorticity origin
(DT)
Core
Near-wall
Core
Near-wall
Positive vorticity
(towards outlet)
Lower half cross-section
Upper half cross-section
Upper half cross-section
Lower half cross-section
Negative vorticity
(away from outlet)
Upper half cross-section
Lower half cross-section
Lower half cross-section
Upper half cross-section
Dimensionless streamwise peak
vorticity -
2
LGDT Inner
1.8
LGDT Outer
2.58x
y = 0.0025e
2
R = 0.99
1.6
RGDT Inner
RGDT Outer
1.4
1.2
1
2.61x
y = 0.0006e
2
R = 0.99
0.8
-0.33x
y = 7.22e
2
R = 0.92
0.6
0.4
-0.49x
y = 7.892e
2
R = 0.95
0.2
0
1.0
2.0
3.0
4.0
5.0
6.0
Dimensionless axial distance (L/D)
7.0
8.0
9.0
Fig. 8: Dimensionless streamwise maximum vorticity as a function of dimensionless axial distance at Re 336
VI. CONCLUSION
We have demonstrated strong evidence of vortex
stretching phenomena in the upstream DT leading up
to the formation of 4-vortex configuration in the
secondary flow of GDT downstream. Scaling
arguments have been used to derive two critical
parameters governing this particular vorticity transport
problem. A simple model for vorticity generation and
transport is proposed, taking into account the
geometric limitations imposed by the rigid walls of the
tubes. Numerical computation of streamwise vorticity
profiles shows reasonable agreement with theory at
low length to diameter ratios. Inaccuracy at higher
length to diameter ratios suggests possible vortex
destretching due to viscous decay.
Acknowledgement
The authors would like to thank Mr Lim Wee Chuan
for suggestions and comments.
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Appendix A: Convective-diffusive transport of
secondary vorticity
boundary condition, the transformed equation (A-8)
becomes
This section entails the solution of eqn (4.2) using
Finite Fourier Transform (FFT). We first nondimensionalize the problem by normalizing the length
variables x with respect to L and z with respect to a,
and the secondary vorticity with respect to (U/a). For
simplicity, we shall use the same notation for the
dimensionless variables in this section as before.
dω yn
The problem statement for ω y ( x, z ) is
∂ω y
∂ 2ω y
dx
+ G (nπ ) ω yn = 0
(A-11)
2
Applying the last boundary condition (A-4), the
solution is then
ω yn = Ω yn e − G (nπ )
2
Ω yn
x
(A-12)
is a constant evaluated as
(A-1)
where
ω y ( x,0) = 0
(A-2)
Ω yn = ω yn (0) = ∫ Φ n ( z )Ω y ( z )dz
ω y ( x,1) = 0
(A-3)
ω y (0, z ) = Ω y ( z )
(A-4)
=G
∂x
∂z
2
1
(A-13)
0
Substitute (A-12) in (A-5) yields the general solution
∞
where G = Re −1 ⎛⎜ L ⎞⎟
a
⎝a⎠
ω y ( x, z ) = 2 ∑ Ω yn e −G (nπ ) x sin nπz
Assuming that the solution can be written in the form
of a series expansion,
Incidentally, if
∞
(A-5)
Φ n ( z ) = 2 sin nπz , n = 1,2,...
where Φ n (z ) is the basis function.
(A-6)
n =1
1
ω yn ( x ) = ∫ Φ n ( z )ω y ( x, z )dz
(A-7)
0
We apply FFT to both sides of (A-1), to give
0
n
⎡ ∂ω y
∂ 2ω y ⎤
−G
( z)⎢
⎥dz = 0
∂z 2 ⎦⎥
⎣⎢ ∂x
(A-8)
The first term is simply
∂ω y
dω yn
1
∫Φ
0
n
( z)
dz =
∂x
(A-9)
dx
Whereas the second term requires integration by parts,
∫
1
0
Φ n ( z)
∂ 2ω y
∂ω y
∂z
2
dz = Φ n
∂ω y
∂z
z =1
z =0
−∫
1
0
dΦ n ∂ω y
dz
dz ∂z
1
⎛ dΦ n
d 2Φ n ⎞
dz ⎟⎟
− ⎜⎜ ω y
− ∫ ωy
0
dz z = 0
∂z z =0 ⎝
dz 2
⎠
∂ω y
⎛
dΦ n ⎞ z =1
⎟ − (nπ )2 ω yn
= ⎜⎜ Φ n
− ωy
dz ⎟⎠ z = 0
∂z
⎝
= Φn
z =1
Ωy
z =1
(A-10)
We have defined our basis function such that
Φ n (0) = Φ n (1) = 0 , and referring to (A-2) and (A-3)
is a constant, (A-14) can be
written as
∞
ω y ( x, z ) = 2Ω y ∑
n =1
The transformed vorticity is defined as
1
(A-14)
n =1
ω y ( x, z ) = ∑ ω yn ( x )Φ n ( z )
∫Φ
2
(1 − (− 1) ) e
n
nπ
−G ( nπ )2 x
sin nπz
(A-15)